Conclusion of Live Challenge

[This post originally appeared on the Brilliant blog on 9/22/2012.]

Thanks for participating in a successful Brilliant Math Fall Challenge. I hope you have benefitted greatly from working on the problems and attending the Master Classes. Winners of the Live Challenge and Week 5 Regional Leaderboards will be announced in early October, once we have verified their results. At that time, we will inform all Live Challenge finalists of their performance.

Brilliant will be back soon with more problems, Master Classes and prizes. In the meantime, we will be posting the Live Challenge problems on the blog. A different Live Challenge will be posted on Monday, Tuesday, Thursday and Friday of each week. We welcome students to submit complete solutions (not just numerical answers) to challenges@brilliantscholars.com beforehand, and will credit your correct submission. You may also participate by commenting on the specific problem post. Comments are currently unmoderated, and can be submitted anonymously. (Note: If your comments do not show up, it has been marked as Spam. One word replies, email addresses and excessive hyperlinks are often marked as Spam.)

The difficulty of the Live Challenge was higher than that of the Weekly Challenges, as there were few trivial points available. The added time constraint also raised the difficulty, as there no longer was the luxury of a whole week to ponder the solution. Students had been told that they were not expected to complete all 20 challenges in 60 minutes. For most students, it was better to have worked on the 'very easy' and 'easy' questions with an obvious approach and accumulated those points. The students that fared better were those that chose to focus on nailing several questions that they knew they could, as opposed to working on all of the problems.

Live Challenge

The following challenges will be posted on Monday and Tuesday in a new blog post. Please post your comments for these questions on the relevant blog posts. Note that students may have received slightly different numerical values for these questions.

Monday: If $\sqrt{79 + 24 \sqrt{7} } = a + b \sqrt{c}$ , where $a, b$ and $c$ are positive integers and $c$ is not divisible by the square of a prime. Determine $a+b+c$ .

Tuesday: Given a cube, a <strong>valid label</strong> is one where each edge is labeled a distinct number from 1 to 12. An <strong>increasing path</strong> is a path formed by the edges labeled $i, j$ and $k$ such that $i < j < k$ and the edges $i, j$ and $k$ form a continuous curve (i.e. edge $j$ has a common vertex with edges $i$ and $k$ , and these 3 edges do not share a common vertex). Over all possible valid labels, what is the minimum number of increasing paths?

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